Trying to replicate a 2D Eye Rig
You can indeed use the boolean modifier to achieve this effect. What you basically need are two cylinders (same shape, scale and location) and two cuboids to replicate the shown behaviour. The intersection between a cuboid and a cylinder will build the shape of the upper or lower eyelid. Add a boolean modifier to each cylinder with one cuboid as target object respectively and set the operation to intersect.




That is basically all you need for the eyelids.

For the eye-shape and pupil just add two new cylinders. To animate the pupil add an empty. Add a Copy Location constraint to the pupil targeting the empty and disable the up-axis of the constraint. Add a Limit Distance constraint and set the target to the cylinder which emulates the eye-shape. Adjust the distance to keep the pupil within the eye. Add a Copy Scale constraint to the pupil targeting the empty so scaling the empty will scale the pupil.



A blend file can be found here.